Estimation of location of discontinuity in a density

Here we propose a few estimators of θ, in addition to those studied in Goria (1978), the point of discontinuity of the probability density $$f(x,\theta ) = \frac{1}{{2\Gamma (\alpha )}}e^{ - |x - \theta |} |x - \theta |^{\alpha - 1} ,$$ for $$0< \alpha< 1, - \infty< x< \infty , - \infty< \theta< \infty .$$ We establish the consistency and the optimality of the Bayes and the maximum probability estimators. Despite their nice properties, these estimators are not easy to compute in this case and their effective computation depends on the knowledge of the exponent α. Hence, we propose another class of estimators, dependent upon the spacings of the observations, computable without actual knowledge of the value of α as long as it is known that α < α₀ < 1: we show that these estimators converge at the best possible rate. We further demonstrate, using a modified version of the maximum probability estimator's technique, that the tails of the density do not substantially effect their efficiency. Finally a bivariate family of densities, having a ridge dependent on the parameter θ, is considered and it is shown that this family exhibits features similar to the univariate case, and thus, the necessary modifications of the arguments of the univariate case are utilized for the estimation of θ in this bivariate example.     

Problem-solving in teaching/learning packages for Technology

In the project Technology in Secondary Education: Problem-solving in Teaching/learning Packages, two experiments regarding a construction problem and an explanation problem were conducted, in which two variants of a teching/learning package (strongly structuredvs. weakly structured) were compared. Only with the strongly structured instructional variant of the package for the explanation problem did the pupils come to a quick solution of the problem. In both experiments, the factors that influenced the problem-solving processes of the pupils were investigated. The results are of importance for curriculum developers and the authors of teaching/learning packages for Technology but further research into the factors influencing the problem solving process of pupils is necessary.